grandes-ecoles 2023 Q17

grandes-ecoles · France · x-ens-maths-b__mp Sequences and series, recurrence and convergence Series convergence and power series analysis

Let $f \in \mathscr { D } _ { \rho } ( \mathbb { R } )$ such that $f ( 0 ) > 0$. Show that there exists $\rho _ { f } \in \mathbb { R } _ { + } ^ { * }$ such that $\rho _ { f } \leqslant \rho$ and such that $f > 0$ on $U _ { \rho _ { f } }$ and $\sqrt { f } \in \mathscr { D } _ { \rho _ { f } } ( \mathbb { R } )$.

Let $f \in \mathscr { D } _ { \rho } ( \mathbb { R } )$ such that $f ( 0 ) > 0$. Show that there exists $\rho _ { f } \in \mathbb { R } _ { + } ^ { * }$ such that $\rho _ { f } \leqslant \rho$ and such that $f > 0$ on $U _ { \rho _ { f } }$ and $\sqrt { f } \in \mathscr { D } _ { \rho _ { f } } ( \mathbb { R } )$.

Paper Questions

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